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<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp; y(x_0)=b_0 \to C_1 y_1(x_0)+C_2 y_2(x_0)+\cdots+C_n y_n(x_0)=b_0,\\
&amp;y^{\prime}(x_0)=b_1 \to C_1 y_1^{\prime}(x_0)+C_2 y_2^{\prime}(x_0)+\cdots+C_n y_n^{\prime}(x_0)=b_1,\\
&amp;\cdots\\
&amp;y^{(n-1)}(x_0)=b_{n-1} \to C_1 y_1^{(n-1)}(x_0)+C_2 y_2^{(n-1)}(x_0)+\cdots+C_n y_n^{(n-1)}(x_0)=b_{n-1}.
\end{aligned}
\end{equation*}
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<p class="continuation">These are the <span class="process-math">\(n\)</span> equations for <span class="process-math">\(C_1, C_2,\cdots, C_n\text{.}\)</span> These equations have unique solution to <span class="process-math">\(C_1, C_2, \cdots, C_n\)</span> if and only if</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
W_0=\left|
\begin{array}{cccc}
y_1(x_0) &amp; y_2(x_0) &amp; \cdots &amp; y_n(x_0)\\
y_1^{\prime}(x_0) &amp; y_2^{\prime}(x_0) &amp; \cdots &amp; y_n^{\prime}(x_0)\\
&amp; &amp; \cdots &amp; \\
y_1^{(n-1)}(x_0) &amp; y_2^{(n-1)} (x_0) &amp; \cdots &amp; y_n^{(n-1)}(x_0)
\end{array}
\right|
\neq 0.
\end{equation*}
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<span class="incontext"><a href="sec4_1.html#p-149" class="internal">in-context</a></span>
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